Question

final exam practice
28 of 49
this quiz: 49 point(s) possible
this question: 1 point(s) possible
solve the following inequality
x² - 9x > 0
...
select the correct choice below and, if necessary, fill in the answer box.
a. the solution set is (-∞,0)∪(9,∞)
(type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b. there is no real solution

Explanation:

Step1: Factor the inequality

Factor \(x^2 - 9x\) to get \(x(x - 9)>0\).

Step2: Find critical points

Set each factor equal to zero: \(x = 0\) and \(x - 9=0\) (so \(x = 9\)). These are the critical points that divide the number line into intervals: \((-\infty, 0)\), \((0, 9)\), and \((9, \infty)\).

Step3: Test intervals

• For \((-\infty, 0)\), pick \(x=-1\): \((-1)(-1 - 9)=(-1)(-10) = 10>0\), so this interval is part of the solution.
• For \((0, 9)\), pick \(x = 1\): \((1)(1 - 9)=(1)(-8)=-8<0\), so this interval is not part of the solution.
• For \((9, \infty)\), pick \(x = 10\): \((10)(10 - 9)=(10)(1)=10>0\), so this interval is part of the solution.

Answer:

A. The solution set is \((-\infty, 0)\cup(9, \infty)\)